aa r X i v : . [ m a t h . A P ] S e p Initial-Boundary Value Problems for ParabolicEquations.
Magnus Fontes
In this paper we prove new existence and uniqueness results for weak solutionsto non-homogeneous initial-boundary value problems for parabolic equationsof the form ∂u∂t − ∇ x · A ( x, t, ∇ x u ) = f in D ′ ( Q + ) (1.1a) u = g on (Ω × { } ) ∪ ( ∂ Ω × R + ) . (1.1b)Here Ω is an open and bounded set in R n and Q + = Ω × R + . Precisestructural conditions for A ( · , · , · ) are given in Section 4, but the model is thefollowing p -parabolic equation ∂u∂t − ∇ x · ( |∇ x u | p − ∇ x u ) = f in D ′ ( Q + ) (1.2a) u = g on (Ω × { } ) ∪ ( ∂ Ω × R + ) , (1.2b)with 1 < p < ∞ .The boundary data is prescribed on the whole parabolic boundary, (Ω ×{ } ) ∪ ( ∂ Ω × R + ), and we study the problem of finding the “largest possible”classes of boundary and source data such that (1.1) has a good meaning andis uniquely solvable.In the case of the elliptic p -laplacian: −∇ · ( |∇ u | p − ∇ u ) = f in D ′ (Ω) (1.3a) u = g on ∂ Ω , (1.3b)1t is well known that W ,p (Ω) is a kind of golden mean. It has the usefulproperty that:Given g ∈ W ,p (Ω), there exists a unique solution u ∈ W ,p (Ω) to the p -laplace equation (1.3) such that u − g belongs to the closure of D (Ω) in the W ,p (Ω)-norm topology. Furthermore the source data ( f in (1.3)) can thenbe taken as sums of first order derivatives of L p/ ( p − (Ω)-functions.In this paper we construct an analogous optimal solution-space for equa-tions of the type (1.1).We point out that our results are new even in the linear case. In thelinear case, where p = 2 and we denote W s, by H s , it is well known (seee.g. [5] Vol. II) that the parabolic solution and lateral boundary valuespaces, replacing the “elliptic spaces” H s (Ω) and H s − / ( ∂ Ω), are H s,s/ (Ω × R + ) and H s − / ,s/ − / ( ∂ Ω × R + ). The initial data on Ω × { } should thenbelong to H s − (Ω) and the natural source data space is H s − ,s/ − (Ω × R + ).With additional compatibility conditions for the coupling of the data in the“corners” of the space-time cylinder we then have unique solvability for thelinear case when s > s = 1, the golden mean inthe elliptic case, several difficulties arise in the parabolic case. One obviousdifficulty is of course that we are in the borderline Sobolev imbedding casein the time direction (half-a-time derivative in L ( R + , L (Ω))), and are thusfor instance unable to define traces on Ω × { } .In Theorem 4.10 we give optimal results in the linear limiting case ( s = 1),and a complete description of the space of solutions (compare with the non-optimal results in e.g. [5],[4] and [3]).We use a similar construction of the solution space (with new technicalcomplications) in the non-linear case when p = 2.Our solution space for a general p , 1 < p < ∞ , (see Definition 4.6) isthe sum of a Banach space carrying initial data and another Banach spacecarrying lateral boundary data. It is a dense subspace of the space of L p ( Q + )-functions, having half order time derivatives in L ( Q + ) and first order spacederivatives in L p ( Q + ).This statement requires some explanation and the appropriate distribu-tion theory, allowing fractional differentiation in the time direction of general L p -functions in a space-time half cylinder, is developed. This analytic frame-work makes it possible to give a precise meaning to the fractional integrationby parts for the time derivatives that is one of the key tools in our method.We point out that we use two different half-a-time derivatives (adjoint to eachother) and that demanding these different derivatives to belong to L ( Q + )2ives rise to different function spaces. In Section 4 we investigate the rela-tions between these different function spaces and discuss some of their basicproperties. It is for instance non-trivial to show that our function spaces arewell behaved when we cut off (in a smooth way) in time. This is, apart fromthe fact that we are in the borderline Sobolev imbedding case in the timedirection, due to the fact that they have non-homogeneous summability andregularity conditions, and that they are defined as spaces of distributions.Most of these technical problems arise already for functions defined on thereal line and half-line, and for clarity we have moved most of these argumentsto an auxiliary section (Section 3) dealing with this case.The main result of this paper is Theorem 4.8 which implies, among otherthings, that our solution space X , / ( Q + ) really is a true analog of the space W ,p (Ω) for the elliptic p -laplacian, in the sense that:Given g ∈ X , / ( Q + ) there exists a unique solution u ∈ X , / ( Q + ) tothe p -parabolic equation (1.1) such that u − g belongs to the closure of D ( Q + )in the X , / ( Q + )-norm topology. Furthermore the source data ( f in (1.1))can be taken as sums of first order space derivatives of L p/ ( p − ( Q + )-functionsand half-a-time derivatives of L ( Q + )-functions.For simplicity we shall assume throughout the paper that the boundaryof Ω is smooth, but this assumption is only used to prove that we can regu-larize functions near the lateral boundary so that the different spaces of testfunctions we use are dense in the corresponding function spaces (see Theorem4.1). We will use the fractional calculus presented in [1]. Here we first give abrief review of the notation and some results. We then extend the calculusto space-time half-cylinders in order to be able to discuss initial-boundaryvalue problems.The Fourier transform on the Schwartz class S ( R n , C ) is defined byˆ u ( ξ ) = Z R n u ( x ) e − i πx · ξ dx, u ∈ S ( R n , C ) . (2.1)The inverse will be denotedˇ u ( ξ ) = Z R n u ( x ) e i πx · ξ dx, u ∈ S ( R n , C ) . (2.2)3he isotropic fractional Sobolev spaces are defined as follows. Definition 2.1
For s ∈ R and < p < ∞ let H sp ( R n , C ) = { u ∈ S ′ ( R n , C ); ((1 + | πξ | ) s/ ˆ u ( ξ )) ∨ ∈ L p ( R n , C ) } . (2.3)They are separable and reflexive Banach spaces with the obvious norms. Wewill use the following multi-index notation. Let α = ( α , . . . , α n ) ∈ R n bean n -tuple. We write α > α j > , j = 1 , . . . , n ; x α = x α · · · x α n n when x ∈ R n ; x α + = x α · · · x α n n + , (where t + = max(0 , t ) for t ∈ R , with asimilar definition for x α − ) and Γ( α ) = Γ( α ) · · · Γ( α n ), where Γ denotes thegamma function. Furthermore we will sometimes write k for the multi-index( k, . . . , k ), the interpretation should be clear from the context. We now definethe classical Riemann-Liouville convolution operators. Definition 2.2
For a multi-index α > , set D − α ± u = χ α − ± ∗ u, u ∈ S ( R n , C ) , (2.4) where the kernels χ α − ± , are given by χ α − ± = Γ( α ) − ( · ) α − ± . (2.5)We extend the definition of D α ± to general multi-indices α ∈ R n in the usualway. Definition 2.3
For α ∈ R n set D α ± u = D k D α − k ± u, u ∈ S ( R n , C ) , (2.6) where we choose the multi-index k ∈ { , , , . . . } n so that k − α > . The definition is independent of the choice of k .Although it is clear in this setting how the support of a function is affectedunder these mappings and also for instance that the operators map realvalued functions to real valued functions, other features become transparenton the Fourier transform side.Computing in S ′ ( R n , C ), we have for all α ∈ R n : D α ± u = ((0 ± i πξ ) α ˆ u ( ξ )) ∨ , u ∈ S ( R n , C ) . (2.7)We will use the following space of test functions.4 efinition 2.4 Let F ( R n , C )= n u ∈ C ∞ ( R n , C ); k u k H sp ( R n , C ) < ∞ , s ∈ R , < p < ∞ o . (2.8) F ( R n , C ) becomes a Fr´echet space with the topology generated by, for in-stance, the following family of semi-norms k · k H sp ( R n , C ) , s ∈ { , , , . . . } ,p = 1 + 2 k , k ∈ Z .We have the following dense continuous imbeddings, D ( R n , C ) ֒ → S ( R n , C ) ֒ → F ( R n , C ) ֒ → E ( R n , C ) . (2.9)An example of a function that belongs to F ( R , C ) but does not belongto S ( R , C ) is x / (1 + x ).For α ≥ D α ± u = ((0 ± i πξ ) α ˆ u ) ∨ , u ∈ F ( R n , C ) . (2.10)The operators D α + and D α − are adjoint to each other and they are connectedthrough the operator H α = n Y k =1 (cos( πα k )Id + sin( πα k ) H k ) , (2.11)where Id is the identity operator and H k is the Hilbert transform with respectto the k th variable, i.e. H k u ( t ) = π − lim ǫ → +0 Z | s |≥ ǫ u ( t − se k ) s ds, u ∈ F ( R n , C ) , (2.12)where e k is the usual canonical k th basis vector in R n . We have the followinglemma. Lemma 2.1
For α ≥ , D α ± are continuous linear operators on F ( R n , C ) .For α ∈ R n , H α is an isomorphism on F ( R n , C ) . For α, β ≥ we have D α ± D β ± = D α + β ± , (2.13) D α + H α = D α − . (2.14) Furthermore all these operators commute on F ( R n , C ) .
5e note that for α ≥ Z R n D α + u Φ dx = Z R n uD α − Φ dx, u, Φ ∈ F ( R n , C ) , (2.15)and for α ∈ R n Z R n H α u Φ dx = Z R n uH − α Φ dx, u, Φ ∈ F ( R n , C ) . (2.16)Now let F ′ ( R n , C ) denote the space of continuous linear functionals on F ( R n , C ), endowed with the weak ∗ topology.Inspired by (2.15) and (2.16), we extend the definition of D α ± and H α to F ′ ( R n , C ) by duality in the obvious way. Definition 2.5
For u ∈ F ′ ( R n , C ) and α ≥ let h D α ± u, Φ i := h u, D α ∓ Φ i , Φ ∈ F ( R n , C ) , (2.17) and for α ∈ R n let h H α u, Φ i := h u, H − α Φ i , Φ ∈ F ( R n , C ) . (2.18)The counterpart of Lemma 2.1 is valid for F ′ ( R n , C ). Lemma 2.2
For α ≥ , D α ± are continuous linear operators on F ′ ( R n , C ) .For α ∈ R n , H α is an isomorphism on F ′ ( R n , C ) . For α, β ≥ we have D α ± D β ± = D α + β ± , (2.19) D α + H α = D α − . (2.20) Furthermore all these operators commute on F ′ ( R n , C ) . We recall that D α ± and H α all take real-valued functions (distributions)to real-valued functions (distributions), and from now on all functions anddistributions will be real valued. We will denote the subspaces of real-valuedfunctions and distributions simply by F ( R n ) and F ′ ( R n ).In [1] we studied parabolic operators on a space-time cylinder Q = Ω × R ,where Ω was a connected and open set in R n . We then introduced thefollowing space of test functions. Definition 2.6
Let F , · ( Q ) denote the subspace of F ( R n × R ) functions withsupport in K × R for some compact subset K ⊂ Ω .
6e put a pseudo-topology on F , · ( Q ) by specifying what sequential con-vergence means. We say that Φ i −→ F , · ( Q ) if and only if the supportsof all Φ i ’s are contained in a fixed set K × R , where K ⊂ Ω is a compactsubset, and k D α Φ i k L P ( Q ) −→ i −→ ∞ for all multi-indices α ∈ Z n +1+ and 1 < p < ∞ .The corresponding space of distributions is then defined as follows. Definition 2.7 If u is a linear functional on F , · ( Q ) , then u is in F ′· , · ( Q ) ifand only if for every compact set K ⊂ Ω , there exist constants C, p , . . . , p N with < p i < ∞ , i = 1 , . . . , N and multi-indices α , . . . , α N with α i ∈ Z n +1+ , i = 1 , . . . , N such that |h u, Φ i| ≤ C N X i =1 k D α i Φ k L pi ( Q ) (2.21) for all Φ ∈ F , · ( Q ) with support in K × R . The motivation for these spaces is that they are invariant under fractionaldifferentiation and Hilbert-transformation in the time variable, and ordinarydifferentiation in the space variables. In the given topologies, these operationsare continuous.For initial-boundary value problems, the parabolic operators will by de-fined on a space-time half-cylinder Q + = Ω × R + , and we shall then needthe following natural spaces of test functions defined on Q + . Remark.
We shall use the same constructions on the real line and half-line, which can be thought of as the case Ω = { } if we identify { } × R with R and { } × R + with R + . Definition 2.8
Let F , · ( Q + ) denote the space of those functions defined on Q + that can be extended to all of Q as elements in F , · ( Q ) .Furthermore let F , ( Q + ) denote the space of those functions defined on Q + that can be extended by zero to all of Q as elements in F , · ( Q ) . (A zero in the first position of course corresponds to zero boundary data onthe lateral boundary and a zero in the second position corresponds to zeroinitial data.)By using the construction in [6] of a (total) extension operator, we seethat F , · ( Q + ) can be identified with the space of all smooth functions Φ,7efined on Q + , with support in K × R + for some compact subset K ⊂ Ω(i.e. they are zero on the complement, with respect to Q + , of K × R + ), with k D α Φ k L P ( Q + ) < ∞ for all multi-indices α ∈ Z n +1+ and 1 < p < ∞ .Thus, we can put an intrinsic pseudo-topology on F , · ( Q + ) by definingthat Φ i −→ F , · ( Q + ) if and only if the supports of all Φ i are contained in afixed set K × R + , where K ⊂ Ω is a compact subset, and k D α Φ i k L P ( Q + ) −→ i −→ ∞ for all multi-indices α ∈ Z n +1+ and 1 < p < ∞ . Then F , ( Q + ) isa closed subspace of F , · ( Q + ) with the induced topology.We also note that D ( Q + ) is densely continuously imbedded in F , ( Q + ).Connected with these spaces of test functions are the following spaces ofdistributions. Definition 2.9 If u is a linear functional on F , · ( Q + ) , then u is in F ′· , ( Q + ) if and only if for every compact set K ⊂ Ω , there exist constants C, p , . . . , p N with < p i < ∞ , i = 1 , . . . , N and multi-indices α , . . . , α N with α i ∈ Z n +1+ , i = 1 , . . . , N such that |h u, Φ i| ≤ C N X i =1 k D α i Φ k L pi (2.22) for all Φ ∈ F , · ( Q + ) with support in K × R + .Furthermore if u is a linear functional on F , ( Q + ) , then u is in F ′· , · ( Q + ) if and only if for every compact set K ⊂ Ω , there exist constants C, p , . . . , p N with < p i < ∞ , i = 1 , . . . , N and multi-indices α , . . . , α N with α i ∈ Z n +1+ , i = 1 , . . . , N such that |h u, Φ i| ≤ C N X i =1 k D α i Φ k L pi ( Q + ) (2.23) for all Φ ∈ F , ( Q + ) with support in K × R + . The importance of these spaces comes from the fact that, for a real-valued α ≥
0, the operations ∂ α + ∂t α := D (0 ,..., ,α )+ : F , ( Q + ) −→ F , ( Q + ) (2.24) ∂ α − ∂t α := D (0 ,..., ,α ) − : F , · ( Q + ) −→ F , · ( Q + ) (2.25)8re continuous. Ordinary differentiations with respect to the space variablesare clearly also continuous operations on these spaces. We shall also use thatthe Hilbert-transform in the time variable h := H (0 ,..., , / : F , ( Q + ) −→ F , · ( Q + ) , (2.26)is a continuous operator.Extending these operators by duality in the obvious way we get that ∂ α + ∂t α : F ′· , ( Q + ) −→ F ′· , ( Q + ) , (2.27) ∂ α − ∂t α : F ′· , · ( Q + ) −→ F ′· , · ( Q + ) , (2.28) h : F ′· , ( Q + ) −→ F ′· , · ( Q + ) , (2.29)and taking ordinary derivatives in the space variables, are continuous oper-ations.Using the total extension operator from [6], one can show that we canidentify F ′· , ( Q + ) with the space of F ′· , · ( Q )-distributions that are zero onΩ × ( −∞ , D ( Q + ) is densely continuously imbedded in F , ( Q + ), we get that F ′· , · ( Q + ) is a continuously imbedded subspace of D ′ ( Q + ).We remark that the space F ′· , ( Q + ) contains elements supported on Ω ×{ } . In fact F ′· , · ( Q + ) ≃ F ′· , ( Q + ) / F ◦ , ( Q + ) , (2.30)where F ◦ , ( Q + ) = { ξ ∈ F ′· , ( Q + ); h ξ, Φ i = 0 , Φ ∈ F , ( Q + ) } .Finally, since F , ( Q + ) is densely continuously imbedded in L p ( Q + ) when1 < p < ∞ , clearly L p ( Q + ) is continuously imbedded in both F ′· , · ( Q + ) and F ′· , ( Q + ) when 1 < p < ∞ . Thus ∂ α + ∂t α : L p ( Q + ) −→ F ′· , ( Q + ) (2.31) ∂ α − ∂t α : L p ( Q + ) −→ F ′· , · ( Q + ) , (2.32)are well-defined continuous operations when 1 < p < ∞ .9 Auxiliary spaces on the real line and half-line.
We shall use the following auxiliary spaces defined on R and in the definition ∂ / − ∂t / should be understood in the F ′ ( R ) distribution sense. Definition 3.1
For < p < ∞ , set B / ( R ) = ( u ∈ L p ( R ); ∂ / − u∂t / ∈ L ( R ) ) . (3.1)We equip these spaces with the following norms. k u k B , / ( R ) := k ∂ / − u∂t / k L ( R ) + k u k L p ( R ) . (3.2)Computing in F ′ ( R ) we see that we can represent these spaces as closedsubspaces of the direct sums L ( R ) ⊕ L p ( R ), and thus they are reflexive andseparable Banach spaces in the topologies arising from the given norms.If { ψ ǫ } is a regularizing sequence it is clear that k ψ ǫ ∗ u k B / ( R ) ≤ k u k B / ( R ) , (3.3)and thus smooth functions are dense in B / ( R ).Due to the definition using distributions and to the inhomogeniety of oursummability conditions, it is unfortunately not so easy to cut off in time andin this way show that F ( R ) (or D ( R )) is dense in B / ( R ). Neverthelessthis is true. Lemma 3.1
The space of testfunctions F ( R ) is dense in B / ( R ) . Proof.
The proof is based on a non-linear version of the Riesz representationtheorem.We (temporarily) denote the closure of F ( R ) in B / ( R ) by B / ( R ), andwe shall show that B / ( R ) = B / ( R ).Set T ( u ) = ∂u∂t + | u | p − u. (3.4)10y fractional integration by parts h T ( u ) , Φ i = Z R ∂ / u∂t / ∂ / − Φ ∂t / + | u | p − u Φ dt ; Φ ∈ F ( R ) , (3.5)and H¨older’s inequality, it is clear that T : B / ( R ) −→ B / ( R ) ∗ . (3.6)is continuous.We notice that T : B / ( R ) −→ B / ( R ) ∗ , (3.7)is weakly continuous and monotone (for definitions see [KS] or [1]).By M. Riesz’ conjugate function theorem, which says that the Hilberttransform h is bounded from L p ( R ) to L p ( R ) (recall that 1 < p < ∞ ), wesee that the operators H α introduced above are isomorphisms on B / ( R ).Now for any α ∈ (0 , /
2) we have h T ( u ) , H − α ( u ) i ≥ Z R sin( πα ) ∂ / u∂t / ∂ / u∂t / (3.8)+(cos( πα ) − sin( πα ) C ) | u | p dt ; u ∈ F ( R ) , (3.9)where C < ∞ is a constant such that k h ( u ) k L p ( R ) ≤ C k u k L p ( R ) . (3.10)Choosing α ∈ (0 , /
2) small enough we see that H α ◦ T is coercive. It followsthat T is a bijection (see [1] for this functional-analytic result and similararguments).Thus given u ∈ B / ( R ) there exists a unique v ∈ B / ( R ) such that T ( u ) = T ( v ) in F ′ ( R ), i.e. ∂ ( u − v ) ∂t + ( | u | p − u − | v | p − v ) = 0 . (3.11)This shows that the difference of elements with the same image has moreregularity in time, namely ∂ ( u − v ) ∂t ∈ L p/ ( p − ( R ).The class of L p ( R ) functions with derivatives in L p/ ( p − ( R ) is stableunder regularization and thus by a continuity argument we see that we can11est with χ ( u − v ), where χ is a cut off function in time, in equation (3.11).We get that (for a canonical continuous representative) t
7→ | u − v | ( t ) isdecreasing. Since u − v belongs to L p ( R ), we conclude that u = v . Thelemma follows. ✷ We are now in position to prove the following lemma.
Lemma 3.2 If u ∈ B / ( R ) then Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt = 2 π Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / − u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt. (3.12) Proof.
Since F ( R ) is dense in B / ( R ) we can compute using the Fouriertransform. Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / − u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z R π | τ || ˆ u | dτ (3.13)= 12 π Z Z R × R | − e i πτs | s | ˆ u ( τ ) | dτ ds. (3.14)Using Parseval’s formula the lemma follows. ✷ We note the following scaling and translation invariance
Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( a ( s − b )) − u ( a ( t − b )) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt = Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt ; a, b ∈ R . (3.15)We also note the following fact. Lemma 3.3
The space B / ( R ) is continuously imbedded in the space offunctions with vanishing mean oscillation, V M O ( R ) . Proof.
Let I ⊂ R denote a bounded interval and let u I denote the meanvalue of u ∈ B / ( R ) over I . Then by Jensen’s inequality1 | I | Z I | u − u I | dt ≤ Z Z I × I (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt. (3.16) ✷ Using the form of the norm in Lemma 3.2, we can now show that we havegood estimates in the B / ( R )-norm for the following cut-off operation.12 emma 3.4 Let χ n be the piecewise affine function that is one on ( − n, n ) ,zero on ( −∞ , − n ) ∪ (2 n, ∞ ) and affine in between. Let I n = ( − n, n ) andfor u ∈ B / ( R ) , denote the mean value of u over I n by u I n . Then thereexists a constant C such that Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) χ n ( u − u I n )( s ) − χ n ( u − u I )( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt ≤ C Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt , (3.17) k χ n ( u − u I n ) k pL p ( R ) ≤ C k u k pL p ( R ) ; u ∈ B / ( R ) . (3.18) Furthermore χ n ( u − u I n ) → u in B / ( R ) as n −→ ∞ . Proof.
The boundedness of the cut-off operation in the L p -norm fol-lows from Jensen’s inequality. For the L -part of the norm an elementarycomputation gives us Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) χ n ( u − u I n )( s ) − χ n ( u − u I )( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt ≤ C ( | I n | Z I n | u − u I n | dt + Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt ) , (3.19)and thus (3.17) follows using (3.16). That χ n u → u in L p ( R ) is clear. If u has compact support, since p >
1, using Jensen’s inequality, we see that χ n u I n → L p ( R ). Since by Jensen’s inequality χ n u I n is uniformly boundedin L p ( R ), a density argument proves that χ n ( u − u I n ) → u in L p ( R ). That χ n ( u − u I n ) → u for the L -part of the norm follows since by an elementarycomputation Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) (1 − χ n )( u − u I n )( s ) − (1 − χ n )( u − u I )( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt (3.20) ≤ C ( | I n | Z I n | u − u I n | dt + Z Z | t | >n (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt ) . (3.21)The last term clearly tends to zero as n tends to infinity. We only have toprove that also 1 | I n | Z I n | u − u I n | dt −→ n → ∞ . This is true since1 | I n | Z I n | u − u I n | dt ≤ n Z Z I n × I n | u ( s ) − u ( t ) | ds dt ≤ C ( log nn Z Z | s | , | t |≤ log n (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt + Z Z | t |≥ log n (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt ) , (3.23)which clearly tends to zero as n tends to infinity. ✷ Remark.
We subtracted the mean value in the argument above in ordernot to have to rely on the fact that u ∈ L p ( R ) when proving boundednessfor the half-derivatives. This is crucial when we later use the same argumenton functions defined in a space-time cylinder. In preparation for this wealso note that, by regularizing, the lemma gives us an explicit sequence of D ( R )-functions tending to a given element in B / ( R ).We now introduce two sets of spaces defined on the real half-line. Definition 3.2
Let B / ( R + ) be the space of functions defined on R + thatcan be extended by zero as elements in B / ( R ) .Furthermore let B / ( R + ) be the space of functions defined on R + thatcan be extended as elements in B / ( R ) . Remark.
The space B / ( R + ) can of course be identified with the closedsubspace of B / ( R ) of functions with support in R + .We now give two simple lemmas, giving intrinsic descriptions of B / ( R + )and B / ( R + ). We omit the proofs, which are straightforward elementarycomputations using the form of the norm in Lemma 3.2. Lemma 3.5
The function space B / ( R + ) is precisely the set of L p ( R + ) -functions such that the following norm is bounded: k u k B / ( R + ) := k u k L p ( R + ) + (cid:26)Z R + u ( t ) t dt + Z Z R + × R + (cid:18) u ( s ) − u ( t ) s − t (cid:19) ds dt ) / . (3.24)14 emma 3.6 The function space B / ( R + ) is precisely the set of L p ( R + ) -functions such that the following norm is bounded: k u k B / ( R + ) := k u k L p ( R + ) + (Z Z R + × R + (cid:18) u ( s ) − u ( t ) s − t (cid:19) ds dt ) / . (3.25) Furthermore, a continuous symmetric extension operator from B / ( R + ) to B / ( R ) is given by E S ( u )( t ) = u ( | t | ) . We have the following density results:
Lemma 3.7
The space F ( R + ) is dense in B / ( R + ) and F ( R + ) is densein B / ( R + ) . Proof.
That F ( R + ) is dense in B / ( R + ) follows immidiately from the factthat F ( R ) is dense in B / ( R ). The argument to prove that F ( R + ) is densein B / ( R + ) is a little more delicate. Given u ∈ B / ( R + ), apriori we onlyknow that there exists a sequence of testfunctions in F ( R ) approaching u inthe B / ( R )-norm.Given u ∈ B / ( R + ) we will show that we can cut-off. Let χ n be thepiecewise affine function that is one on (0 , n ), zero on (2 n, ∞ ) and affine inbetween. We will show that χ n u −→ u in B / ( R + ). Taking this for grantedwe can regularize with a regularizing sequence having support in R + whichgives us the lemma.That χ n u −→ u in L p ( R + ) is clear. We now estimate the L -part of thenorm. An elementary computation gives us Z R + ((1 − χ n ) u ) ( t ) t dt + Z Z R + × R + ((1 − χ n ) u ( s ) − (1 − χ n ) u ( t )) ( s − t ) ds dt ≤ C ( n Z n u ( t ) dt + Z Z ( n, ∞ ) × R + (cid:18) u ( s ) − u ( t ) s − t (cid:19) ds dt + Z ∞ n u ( t ) t dt (cid:27) . (3.26)The last two terms above clearly tend to zero as n → ∞ . To estimate thefirst term, we integrate by parts (we may assume that u is smooth, it is the15ecay at infinity that is the issue).12 n Z n u ( t ) dt = 12 n Z n (cid:18)Z n u ( s ) s ds − Z t u ( s ) s ds (cid:19) dt ≤ n Z n log n Z nt u ( s ) s ds dt + log n n Z n u ( s ) s ds, (3.27)which clearly tends to zero as n tends to infinity. The lemma follows. ✷ We now give the following equivalent characterization of B / ( R + ). Lemma 3.8
A function u ∈ L p ( R + ) belongs to B / ( R + ) if and only if the F ′ ( R + ) -distribution derivative ∂ / u∂t / ∈ L ( R + ) . Furthermore an equivalentnorm on B / ( R + ) is given by k u k = k u k L p ( R + ) + k ∂ / u∂t / k L ( R + ) . (3.28) Remark.
We recall that the F ′ ( R + )-distribution derivative, apart fromwhat happens inside R + , also controls what happens on the boundary { } .The fact that ∂ / u∂t / ∈ L ( R + ) thus actually contains a lot of informationabout u ’s behaviour at 0. Proof.
It is clear that a function in B / ( R + ) has the F ′ ( R + )-distribution deriva-tive ∂ / u∂t / in L ( R + ).On the other hand, let E be the extension by zero operator. Then if u ∈ L p ( R + ) and the F ′ ( R + )-distribution derivative ∂ / u∂t / ∈ L ( R + ) we have Z R E ( u ) ∂ / − Φ ∂t / dt = Z R E ( ∂ / u∂t / )Φ dt ; Φ ∈ F ( R ) . (3.29)This shows that the F ′ ( R )-distribution derivative ∂ / E ( u ) ∂t / belongs to L ( R ).An easy computation shows that Z R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / E ( u )( t ) ∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ∼ Z Z R + × R + (cid:12)(cid:12)(cid:12)(cid:12) E ( u )( s ) − E ( u )( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt + Z R + E ( u ) t dt. (3.30)16ince E ( u ) = u on (0 , ∞ ) the lemma follows. ✷ We now give a corresponding equivalent norm on B / ( R + ). Lemma 3.9 If u ∈ B / ( R + ) , then the F ′ ( R + ) -distribution derivative ∂ / − u∂t / belongs to L ( R + ) .Furthermore an equivalent norm on B / ( R + ) is given by k u k = k u k L p ( R ) + k ∂ / − u∂t / k L ( R + ) . (3.31) Remark.
In contrast to the F ′ ( R + )-distribution derivative, the F ′ ( R + )-distribution derivative that we use in this definition “does not see” whathappens on the boundary, { } . Proof.
Since F ( R + ) is dense in B / ( R + ), it is enough to show that Z R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / − u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ∼ Z Z R + × R + (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt, (3.32)for functions in F ( R + ), where ∼ means that the seminorms are equivalent.For p = 2 we (temporarily) denote the closure of F ( R + ) in the norm k u k = k ∂ / − u∂t / k L ( R + ) + k u k L ( R + ) , (3.33)by H .It follows directely from the definitions, and the fact that F ( R + ) is densein B / ( R + ), that B / ( R + ) is continuously imbedded in H .We shall now show that in fact H = B / ( R + ).Let T denote the operator T : u ∂u∂t + u . Then T : B / ( R + ) −→ H ∗ iscontinuous. This follows from fractional integration by parts, h T u, Φ i = ∂ / u∂t / , ∂ / − Φ ∂t / ! L + ( u, Φ) L ; Φ ∈ F ( R + ) , u ∈ F ( R + ) , (3.34)and the fact that F ( R + ) is dense in H and that F ( R + ) is dense in B / ( R + ).17ow by the Hahn-Banach theorem, given ξ ∈ H ∗ there exist elements u, v ∈ L ( R + ) such that h ξ, Φ i = u, ∂ / − Φ ∂t / ! L + ( v, Φ) L ; Φ ∈ F ( R + ) . (3.35)We can thus extend ξ by zero to an element E ( ξ ) of B / ( R ) ∗ . Since T : B / ( R ) → B / ( R ) ∗ is an isomorphism, we can find a unique element u ∈ B / ( R ) such that T u = E ( ξ ) in F ′ ( R ). But this holds if and only if u ∈ B / ( R + ) and T u = ξ in F ′ ( R + ).Thus T : B / ( R + ) −→ H ∗ is an isomorphism.Furthermore, by direct computation (or by interpolation (recall that p =2)), we know that T : B / ( R + ) −→ B / ( R + ) ∗ (3.36)is an isomorphism.Since F ( R + ) is densely continuously imbedded in both H and B / ( R + )and thus H ∗ and B / ( R + ) ∗ both are well defined subspaces in F ′ ( R + ),we see that H ∗ and B / ( R + ) ∗ are identical as subspaces of F ′ ( R + ) andequivalent as Hilbert spaces.Since B / ( R + ) ֒ → H , by Riesz representation theorem, this implies that H and B / ( R + ) have equivalent norms.From a scaling argument it now follows that Z R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / − u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ∼ Z Z R + × R + (cid:12)(cid:12)(cid:12)(cid:12) u ( s ) − u ( t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt, (3.37)for functions in B / ( R + ). The lemma follows. ✷ We shall consider operators of the form
T u = ∂u∂t − ∇ x · A ( x, t, ∇ x u ) , (4.1)on a space-time cylinder Q + = Ω × R + , where Ω is an open and bounded setin R n with smooth boundary.We shall assume the following structural conditions for the function A :Ω × R + × R n −→ R n . 18. Q + ∋ ( x, t ) A ( x, t, ξ ) is Lebesgue measurable for every fixed ξ ∈ R n .2. R n ∋ ξ A ( x, t, ξ ) is continuous for almost every ( x, t ) ∈ Q + .3. For every ξ, η ∈ R n , ξ = η and almost every ( x, t ) ∈ Q + , we have( A ( x, t, ξ ) − A ( x, t, η ) , ξ − η ) > . (4.2)4. There exists p ∈ (1 , ∞ ), a constant λ > h ∈ L ( Q + )such that for every ξ ∈ R n and almost every( x, t ) ∈ Q + :( A ( x, t, ξ ) , ξ ) ≥ λ | ξ | p − h ( x, t ) . (4.3)5. There exists a constant Λ ≥ λ > H ∈ L p/ ( p − ( Q + )such that for every ξ ∈ R n and almost every ( x, t ) ∈ Q + : | A ( x, t, ξ ) | ≤ Λ | ξ | p − + H ( x, t ) . (4.4)The Carath´eodory conditions 1 and 2 above guarantee that the function Q ∋ ( x, t ) A ( x, t, Φ( x, t )) is measurable for every function Φ ∈ L p ( Q + , R n ).Condition 3 is a strict monotonicity condition that gives us uniqueness re-sults. Conditions 4 (coercivity) and 5 (boundedness) give us apriori estimatesthat imply existence results (see [1]).We now introduce some function spaces, and in their definitions ∂ / − /∂t / should be understood in the F ′· , · ( Q ) distribution-sense. Definition 4.1
For < p < ∞ , set B , / · , · ( Q ) = ( u ∈ L p ( Q ); ∂ / − u∂t / ∈ L ( Q ) , ∂u∂x i ∈ L p ( Q ) , i = 1 , . . . , n. (cid:27) . (4.5)We equip these spaces with the following norms. k u k B , / · , · ( Q ) = k ∂ / − u∂t / k L ( Q ) + k u k L p ( Q ) + n X i =1 k ∂u∂x i k L p ( Q ) . (4.6)Computing in F ′· , · ( Q ) we see that we can represent these spaces as closedsubspaces of the direct sum L ( Q ) ⊕ L p ( Q ) ⊕ · · · ⊕ L p ( Q ), and thus they19re reflexive and separable Banach spaces in the topologies arising from thegiven norms.Since the lateral boundary is smooth (in fact Lipschitz continuous suf-fices), we can extend an element in B , / · , · ( Q ) to all of R n × R and then cutoff in the space variables. By regularizing it is clear that functions smoothup to the boundary are dense in B , / · , · ( Q ). To show that F · , · ( Q ) is densein B , / · , · ( Q ) we only have to prove that we can “cut off” in time. This willfollow as in Lemma 3.4 once we have the following result. Lemma 4.1 If u ∈ B , / · , · ( Q ) , then Z Z Z Ω × R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( x, s ) − u ( x, t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) dx ds dt = 2 π Z Z Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / − u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx dt. (4.7) Proof.
That ∂ / − u∂t / = v means that Z Z Q u ( x, t ) ∂ / Φ( x, t ) ∂t / dx dt = Z Z Q v ( x, t )Φ( x, t ) dx dt ; Φ ∈ F , · ( Q ) . (4.8)Now for almost every x ∈ Ω, Ω ∋ x u ( x, · ) ∈ L p ( R ) and Ω ∋ x v ( x, · ) ∈ L ( R ) are well defined. Let S denote the set of common Lebesgue points.Since the Lebesgue points of a function can only increase by multiplicationwith a smooth function, by taking limits of mean values, we get that Z R u ( x, t ) ∂ / Φ( x, t ) ∂t / dt = Z R v ( x, t )Φ( x, t ) dt ; Φ ∈ F , · ( Q ) , (4.9)for all x ∈ S . This implies that for almost every x ∈ Ω the L p ( R ) function t u ( x, t ) has half a derivative equal to v ( x, t ) ∈ L ( R ). So from theone-dimensional result it follows that Z Z R × R (cid:12)(cid:12)(cid:12)(cid:12) u ( x, s ) − u ( x, t ) s − t (cid:12)(cid:12)(cid:12)(cid:12) ds dt = 2 π Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ / − u∂t / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt, (4.10)for almost every x ∈ Ω. Integrating with respect to x , the lemma follows. ✷ We conclude that: 20 emma 4.2
The space of testfunctions F · , · ( Q ) is dense in B , / · , · ( Q ) . We now introduce the following subspace that corresponds to zero bound-ary data on the lateral boundary ∂ Ω × R and as | t | → ∞ . Definition 4.2
Let B , / , · ( Q ) denote the closure of F , · ( Q ) in the B , / · , · ( Q ) -topology. We shall work with the following two sets of function spaces on Q + . Definition 4.3
Let B , / ∗ , · ( Q + ) denote the space of functions defined on Q + that can be extended to elements in B , / ∗ , · ( Q ) .Furthermore let B , / ∗ , ( Q + ) denote the space of functions defined on Q + that can be extended by zero to elements in B , / ∗ , · ( Q ) . Here ∗ optionally stands for · or 0. A zero in the first position corresponds tozero boundary data on the lateral boundary and a zero in the second positioncorresponds to zero initial data.Clearly B , / ∗ , ( Q + ) can be identified with a closed subspace of B , / ∗ , · ( Q ).We give the following two simple lemmas concerning these spaces and, asin the case of the real line, we omit the easy proofs. Lemma 4.3
The function space B , / ∗ , ( Q + ) becomes a Banach space withthe norm k u k B , / ∗ , ( Q + ) = k u k L p ( Q + ) + k∇ x u k L p ( Q + ) + (cid:26)Z Q + u ( x, t ) t dt dx + Z Z Z Ω × R + × R + (cid:18) u ( x, s ) − u ( x, t ) s − t (cid:19) dx ds dt ) / . (4.11) Lemma 4.4
The function space B , / ∗ , · ( Q + ) becomes a Banach space withthe norm k u k B , / · , · ( Q + ) = k u k L p ( Q + ) + k∇ x u k L p ( Q + ) + (Z Z Z Ω × R + × R + (cid:18) u ( x, s ) − u ( x, t ) s − t (cid:19) dx ds dt ) / . (4.12) Furthermore a continuous symmetric extension mapping from B , / ∗ , · ( Q + ) to B , / ∗ , · ( Q ) is given by E S ( u )( x, t ) = u ( x, | t | ) . F ′· , ( Q + ) we can give an equivalent characterization of B , / · , ( Q + ). Lemma 4.5
A function u ∈ L p ( Q + ) belongs to B , / · , ( Q + ) if and only if the F ′· , ( Q + ) -distribution derivative ∂ / u∂t / belongs to L ( Q + ) , and the F ′· , · ( Q + ) -distribution derivatives ∇ x u ∈ L p ( Q + ) . Furthermore an equivalent norm on B , / · , ( R + ) is then given by k u k = k∇ x u k L p ( Q + ) + k u k L p ( Q + ) + k ∂ / u∂t / k L ( Q + ) . (4.13) Proof.
As on the real line. ✷ Using the corresponding result on the real half-line and the same type ofargument as in the proof of Lemma 4.1, we see that an equivalent norm on B , / ∗ , · ( Q + ) is given by k u k = k ∂ / − u∂t / k L ( Q + ) + k u k L p ( Q + ) + n X i =1 k ∂u∂x i k L p ( Q + ) , (4.14)where ∂ / − ∂t / is understood in the F ′· , · ( Q + )-distribution sense.We have the following density results: Theorem 4.1
The space of testfunctions F · , ∗ ( Q + ) is dense in B , / · , ∗ ( Q + ) .Furthermore the space of testfunctions F , ∗ ( Q + ) is dense in B , / , ∗ ( Q + ) . Proof.
Since the boundary of Ω is smooth we have good extension opera-tors in the space variables, and we can also translate the support of functionsaway from the lateral boundary without spreading the support in the timedirection. The result thus follows exactly as in Lemma 3.7. ✷ We point out the following result that follows immediately from the givennorms.
Lemma 4.6
The space B , / ∗ , ( Q + ) is continuously imbedded in B , / ∗ , · ( Q + ) . We also remark that the (semi)norms k ∂ / − u∂t k L ( Q + ) and k ∂ / u∂t k L ( Q + ) arenot equivalent. In fact in Lemma 4.8 below we show that B , / , ( Q + ) is adense subspace of B , / , · ( Q + ). This is of course connected with the well22nown fact that if u ∈ L ( Q ) and ∂ / − u∂t / ∈ L ( Q ), it is in general impossibleto define a trace on Ω × { } (for instance the function ( x, t ) log | log | t || locally belongs to this space). Still a function in B , / · , ( Q + ) is of course zeroon Ω × { } in the sense that Z Z Q + u ( x, t ) t dxdt < ∞ . (4.15)We shall now discuss homogeneous data on the whole parabolic boundary. We introduce the following space of F ′· , · ( Q )-distributions defined globally intime, but supported in Q + . Definition 4.4
Let B − , − / · , ( Q + ) := n ξ ∈ B , / , · ( Q ) ∗ ; ξ = 0 in Ω × ( −∞ , o (4.16)From Theorem 4.3 and Theorem 4.4 in [1] follows Theorem 4.2
For T as defined in (4.1), satisfying the structural conditions(1)–(5), T : B , / , ( Q + ) −→ B − , − / · , ( Q + ) (4.17) is a bijection. We shall now show that B − , − / · , ( Q + ) can be identified with the dualspace of B , / , · ( Q + ). Lemma 4.7
We can identify B − , − / · , ( Q + ) with B , / , · ( Q + ) ∗ . Remark.
Note that we here identify a subspace of F ′· , · ( Q ) with a subspaceof F ′· , ( Q + ). Proof.
Given ξ ∈ B , / , · ( Q + ) ∗ we have (by the Hahn-Banach theorem) u ∈ L ( Q + ) and u i ∈ L p ′ ( Q + ), i = 1 , . . . , n such that h ξ, Φ i = Z Z Q + u ∂ / − Φ ∂t + n X i =1 u i ∂ Φ ∂x i dxdt ; Φ ∈ F , · ( Q + ) . (4.18)23t is thus clear that we can extend this ξ to all of F , · ( Q ) by zero. Set h ξ , Φ i = Z Z Q E ( u ) ∂ / − Φ ∂t + n X i =1 E ( u i ) ∂ Φ ∂x i dxdt ; Φ ∈ F , · ( Q ) , (4.19)where E denotes the operator that extends a function with 0 to all of Q .The mapping B , / , · ( Q + ) ∗ ∋ ξ ξ ∈ B − , − / · , ( Q + ) is clearly injective, butit is also surjective. This follows since given ξ ∈ B − , − / · , ( Q + ), by Theorem4.2 above, there exists a (unique) u ξ ∈ B , / , ( Q + ) such that ∂u ξ ∂t − ∇ x · ( |∇ x u ξ | p − ∇ x u ξ ) = ξ, (4.20)i.e. h ξ, Φ i = Z Z Q ∂ / u ξ ∂t ∂ / − Φ ∂t + ( |∇ x u ξ | p − ∇ x u ξ ) · ∇ x Φ dxdt ; Φ ∈ F , · ( Q ) , (4.21)and we see that ξ has the required form. ✷ Thus we can reformulate Theorem 4.2.
Theorem 4.3
For T as defined in (4.1), satisfying the structural conditions(1)–(5), T : B , / , ( Q + ) −→ B , / , · ( Q + ) ∗ (4.22) is a bijection. Remark.
This theorem of course means that given ξ ∈ B , / , · ( Q + ) ∗ thereexists a unique u ∈ B , / , ( Q + ) such that h T ( u ) , Φ i = h ξ, Φ i ; Φ ∈ B , / , · ( Q + ) . (4.23)Which means precisely that h ξ, Φ i = Z Z Q + ∂ / u ξ ∂t ∂ / − Φ ∂t + A ( x, t, ∇ x u ) · ∇ x Φ dxdt ; Φ ∈ F , · ( Q + ) , (4.24)since F , · ( Q + ) is dense in B , / , · ( Q + ).The following structure theorem for our source data space is an immediateconsequence of the Hahn-Banach theorem.24 heorem 4.4 Given ξ ∈ B , / , · ( Q + ) ∗ there exist functions u ∈ L ( Q + ) and u , . . . , u n ∈ L p/ ( p − ( Q + ) such that ξ = ∂ / u ∂t + n X i =1 ∂u i ∂x i (4.25) in F ′· , ( Q + ) . Our next result implies that in general it is actually enough to test ourequations with F , ( Q + ) instead of F , · ( Q + ). Lemma 4.8
The continuous imbedding B , / , ( Q + ) ֒ → B , / , · ( Q + ) (4.26) is dense. Proof.
It is enough to show that if ξ ∈ B , / , · ( Q + ) ∗ and h ξ, Φ i = 0 for allΦ ∈ B , / , ( Q + ), then ξ = 0.Now given ξ ∈ B , / , · ( Q + ) ∗ , by Theorem 4.3, there exists a unique u ξ ∈ B , / , ( Q + ) such that ∂u ξ ∂t − ∇ x · ( |∇ x u ξ | p − ∇ x u ξ ) = ξ. (4.27)Now if h ξ, Φ i = 0 for all Φ ∈ B , / , ( Q + ), then with Φ = u ξ we get Z Z Q + |∇ x u ξ | p dx dt = 0 . (4.28)By the Poincar´e inequality u ξ = 0, and so ξ = 0. ✷ We will first introduce the space that will carry the initial data. In thedefinition, all derivatives should be understood in the F ′· , · ( Q + )-distributionsense. 25 efinition 4.5 Let B I ( Q + ) = n u ∈ B , / , · ( Q + ) ∩ C b ([0 , ∞ ) , L (Ω)); ∂u∂t ∈ L p ′ ( R + , W − ,p ′ (Ω)) (cid:27) . (4.29)Here C b ([0 , ∞ ) , L (Ω)) denotes the space of bounded continuous functionsfrom [0 , ∞ ) into L (Ω), and ∂u∂t ∈ L p ′ ( R + , W − ,p ′ (Ω)) means exactly that |h u, ∂ Φ ∂t i| ≤ C k∇ x Φ k L p ( Q + ) ; Φ ∈ F , ( Q + ) , (4.30)for some constant C >
0. The smallest possible constant is by definition k ∂u∂t k L p ′ ( R + ,W − ,p ′ (Ω)) .We equip B I ( Q + ) with the following norm k u k B I ( Q + ) := k u k B , / , · ( Q + ) + sup t ∈ R + k u ( · , t ) k L (Ω) + k ∂u∂t k L p ′ ( R + ,W − ,p ′ (Ω)) . (4.31)Using Theorem 4.3 and the monotonicity of A ( x, t, · ) we shall now provethat we always have a unique solution in B I ( Q + ) to the following initial valueproblem. Theorem 4.5
Given u ∈ L (Ω) , there exists a unique element u ∈ B I ( Q + ) such that ∂u∂t − ∇ x · A ( x, t, ∇ x u ) = 0 in F ′· , · ( Q + ) (4.32a) u = u on Ω × { } . (4.32b) Proof.
Uniqueness follows immediately from the monotonicity of A ( x, t, · ) bypairing with a cut off function in time multiplied with the difference of twosolutions. To prove existence we first note that if u ∈ D (Ω), we can extend itfor instance to a smooth testfunction U ∈ D (Ω × ( − , U ( x, t ) = u ( x ) when − < t < ∂U ∂t ∈ B , / , · ( Q + ) ∗ , by Theorem 4.3, we know that there exists aunique w ∈ B , / , ( Q + ) such that ∂w∂t − ∇ x · A ( x, t, ∇ x w + ∇ x U ) = − ∂U ∂t in B , / , · ( Q + ) ∗ . (4.33)26hen clearly u = ( w + U ) ∈ B , / , · ( Q + ) solves (4.32), and the initial valueis taken in the sense that Z Z Ω × (0 , ( u ( x, t ) − u ( x )) t dx dt < ∞ . (4.34)By standard arguments it follows from (4.32) that u ∈ B I ( Q + ) and so theinitial data is actually taken in C b ([0 , ∞ ) , L (Ω))-sense.Given u ∈ L (Ω) we now choose a sequence D (Ω) ∋ u n −→ u in L (Ω).Let u n denote the solution of (4.32) with initial data u n . By testing with u n χ , where χ is a standard cut off function in time, in (4.32), we get thatsup t ∈ R + Z Ω ( u n − u m ) ( x, t ) dx ≤ Z Ω ( u n − u m ) ( x ) dx. (4.35)It is also clear that k∇ x u n k L p ( Q + ) is bounded by a constant independent of n . Finally we note that we can extend u n symmetrically to Q and the ex-tended function E S ( u n ) ∈ B , / , · ( Q ) will satisfy ∂E S ( u n ) ∂t ∈ L p ′ ( R , W − ,p ′ (Ω)).We then have Z Z Q ∂ / − E S ( u n ) ∂t / ∂ / − Φ k ∂t / dx dt = Z R h ∂E S ( u n ) ∂t , h (Φ k ) i dt, (4.36)for a sequence F , · ( Q ) ∋ Φ k → E S ( u n ) in B , / , · ( Q ).This implies that k ∂ / − E S ( u n ) ∂t / k L ( Q + ) is bounded by a constant independentof n .We conclude that k u n k B I ( Q + ) ≤ C , where C < ∞ is a constant indepen-dent of n .We can now extract a weakly convergent subsequence and in fact, as wehave seen, we actually have strong convergence in C b ([0 , ∞ ) , L (Ω)) and thusthe limit function satisfies the initial conditions.Finally a Minty argument using the monotonicity of A ( x, t, · ) shows thatthe limit function solves (4.32). The theorem follows. ✷ We shall now introduce the function space that will carry both initial andlateral boundary data. 27ince we have continuous imbeddings B , / , · ( Q + ) ֒ → B , / · , · ( Q + ) and B I ( Q + ) ֒ → B , / · , · ( Q + ), the following definition makes sense. Definition 4.6
Let X , / ( Q + ) = B , / · , ( Q + ) + B I ( Q + ) , (4.37) be equipped with the norm k u k X , / ( Q + ) = inf ( u ,u ) ∈ K u (cid:16) k u k B , / · , ( Q + ) + k u k B I ( Q + ) (cid:17) , (4.38) where the infimum is taken over the set K u = n ( u , u ); u + u = u, u ∈ B , / · , ( Q + ) , u ∈ B I ( Q + ) o . (4.39)The following imbeddings are immediate k u k X , / ( Q + ) ≤ k u k B , / · , ( Q + ) ; u ∈ B , / · , ( Q + ) , (4.40) k u k X , / ( Q + ) ≤ k u k B I ( Q + ) ; u ∈ B I ( Q + ) , (4.41) k u k B , / · , · ( Q + ) ≤ C k u k X , / ( Q + ) ; u ∈ X , / ( Q + ) . (4.42)For an element in X , / ( Q + ) we can always define the trace on Ω × { } . Theorem 4.6
There exists a continuous linear and surjective trace operator
T r : X , / ( Q + ) −→ L (Ω) . (4.43) There also exists a bounded extension operator E : L (Ω) −→ X , / ( Q + ) (4.44) such that T r ◦ E = Id L (Ω) . Proof.
Given u ∈ X , / ( Q + ), there exist u ∈ B , / · , ( Q + ) and u ∈ B I ( Q + )such that u = u + u . Since u ∈ B I ( Q + ) = ⇒ u ∈ C b ([0 , + ∞ ) , L (Ω)), u | Ω ×{ } is a well defined element of L (Ω). We now define u | Ω ×{ } = u | Ω ×{ } .We have to show that this is independent of the decomposition of u , but28f we have two different decompositions u + u = v + v as above, then( u − v ) ∈ B I ( Q + ) ∩ B , / · , ( Q + ), which implies that Z Z Ω × (0 , + ∞ ) ( u − v ) ( x, t ) t dxdt < + ∞ , (4.45)and so u ( · ,
0) = v ( · ,
0) since they both belong to C b ([0 , + ∞ ) , L (Ω)).Now k u ( · , k L (Ω) = k u ( · , k L (Ω) ≤ C k u k B I ( Q + ) , (4.46)for any decomposition u = u + u as above. Taking the infimum over allsuch decompositions gives: k u ( · , k L (Ω) ≤ C k u k X , / ( Q + ) , u ∈ X , / ( Q + ) . (4.47)Now given u ∈ L (Ω), let E ( u ) be the (unique) solution in B I ( Q + ) ofthe initial value problem: ∂u∂t − ∇ x · ( |∇ x u | p − ∇ x u ) = 0 in Q + = Ω × R + (4.48a) u = u on Ω × { } . (4.48b)Clearly this extension map satisfies T r ◦ E = Id L (Ω) and furthermore k E ( u ) k B I ( Q + ) ≤ C k u k L (Ω) , (4.49)and thus k E ( u ) k X , / ( Q + ) ≤ C k u k L (Ω) . (4.50) ✷ Remark.
Note that if p = 2 the extension map is linear. Theorem 4.7
We have the following imbedding: k u k B , / · , ( Q + ) ≤ C k u k X , / ( Q + ) ; u ∈ B , / · , ( Q + ) . (4.51) Proof. If u ∈ B , / · , ( Q + ), and u = u + u with u ∈ B , / · , ( Q + ) and u ∈ B I ( Q + ), then u ( · ,
0) = 0 since u ∈ B , / · , ( Q + ) ∩ B I ( Q + ). Thus u canbe extended by zero to all of Q . Since, by a continuity argument, k ∂ / u ∂t k L ( Q + ) = − Z R + h ∂u ∂t , h ( u ) i dt, u ∈ B , / · , ( Q + ) ∩ B I ( Q + ) . (4.52)29e get k u k B , / · , ( Q + ) + k u k B I ( Q + ) ≥ C (cid:16) k u k B , / · , ( Q + ) + k u k B , / · , ( Q + ) (cid:17) ≥ C k u + u k B , / · , ( Q + ) = C k u k B , / · , ( Q + ) , (4.53)where C >
0. Taking the infimum concludes the proof. ✷ We immediatelyget the following
Corollary 4.1
There exist constants C , C > such that C k u k B , / , ( Q + ) ≤ k u k X , / ( Q + ) ≤ C k u k B , / , ( Q + ) ; u ∈ F , ( Q + ) . (4.54) Thus B , / , ( Q + ) is the closure of F , ( Q + ) in the X , / ( Q + ) -norm topology. We are now ready to state our main theorem.
Theorem 4.8
Given f ∈ B , / , · ( Q + ) ∗ and g ∈ X , / ( Q + ) , there exists aunique element u ∈ X , / ( Q + ) such that ∂u∂t − ∇ x · ( A ( x, t, ∇ x u )) = f in F ′· , · ( Q + ) (4.55a) u − g ∈ B , / , ( Q + ) . (4.55b) Proof.
Let w = u − g . Then (4.55) is equivalent to ∂w∂t − ∇ x · ( A ( x, t, ∇ x ( w + g ))) = f − ∂g∂t in F ′· , · ( Q + ) (4.56a) w ∈ B , / , ( Q + ) . (4.56b)Here ∂g∂t ∈ F ′· , · ( Q + ) has a unique extension to an element in B , / , · ( Q + ) ∗ . Infact, if g ∈ X , / ( Q + ), we can write g = g + g , where g ∈ B , / · , ( Q + ) and g ∈ B I ( Q + ). Thus |h g, ∂ Φ ∂t i| = |h g , ∂ Φ ∂t i + h g , ∂ Φ ∂t i|≤ C (cid:16) k g k B , / · , ( Q + ) + k g k B I ( Q + ) (cid:17) k Φ k B , / , · ( Q + ) ; Φ ∈ F , ( Q + ) . (4.57)30ince, by Lemma 4.8 and Theorem 4.1, F , ( Q + ) is dense in B , / , · ( Q + ), itis clear that we have a unique extension. If the function A ( · , · , · ) satisfiesthe structural conditions 1–5 given above, then also A ( · , · , · + g ), with g ∈ X , / ( Q + ), satisfies the same structural conditions (with new constants λ, Λand functions
H, h depending on g ). Thus Theorem 4.3, and the remarkfollowing Theorem 4.3, tell us that (4.56) has a unique solution. This impliesthat u = w + g is the unique solution to (4.55). ✷ Remark.
Note that since D ( Q + ) is densely continuously imbedded in F , ( Q + ) it is equivalent to demand that (4.55) should hold in D ′ ( Q + ).We shall conclude with a comment on the linear case.The function spaces we have introduced so far coincides with well knownfunction spaces existing in the literature when p = 2. When p = 2 we shallfollow existing notation and replace B with H for all spaces (for instance if p = 2 we shall write H , / , · ( Q + ) instead of B , / , · ( Q + ) and so on).The Sobolev space H / , / · , · ( ∂ Ω × R + ) below is defined by pull-backs inlocal charts on ∂ Ω. Theorem 4.9 If p = 2 there exists a linear, continuous and surjective traceoperator T r : X , / ( Q + ) −→ H / , / · , · ( ∂ Ω × R + ) . (4.58) There also exists a continuous and linear extension operator E : H / , / · , · ( ∂ Ω × R + ) −→ X , / ( Q + ) , (4.59) such that T r ◦ E = Id | H / , / · , · ( ∂ Ω × R ) . Proof.
Using a partition of unity argument and the Fourier multiplier op-erators m s ( D ) u = ((1 + i πτ + 4 π | ξ | ) − s ˆ u ) ∨ ; s ∈ R , (4.60)which preserves forward support in time, and have the property that m s ( D ) (cid:0) L ( R n × R ) (cid:1) = H s,s · , · ( R n × R ) , (4.61)we can construct continuous linear operators: T r : H , / · , ( Q + ) −→ H / , / · , · ( ∂ Ω × R + ) (4.62)and E : H / , / · , · ( ∂ Ω × R + ) −→ H , / · , ( Q + ) , (4.63)31uch that T r ◦ E = Id | H / , / · , · ( ∂ Ω × R ) . Now given u ∈ X , / ( Q + ), let u = u + u where u ∈ H , / · , ( Q + ) and u ∈ H I ( Q + ). We define u | ∂ Ω × R + = u | ∂ Ω × R + .This definition is independent of the decomposition of u . In fact, if u + u = v + v are two decompositions as above, then u − v ∈ L ( R + , H (Ω)), andso ( u − v ) | ∂ Ω × R = 0.Now k T r ( u ) k H / , / · , · ( ∂ Ω × R + ) ≤ C k u k H , / · , ( Q + ) , (4.64)for any decomposition. Taking the infimum proves the continuity of T r .The continuity of the extension operator E follows from the imbedding H , / · , ( Q + ) ֒ → X , / ( Q + ). ✷ Combining our trace theorems with Theorem 4.8 gives us in the linearcase:
Theorem 4.10 If T u = ∂u∂t − ∇ x · ( A ( x, t, ∇ x u )) , (4.65) is a linear operator, satisfying the structural conditions 1–5 above, then X , / ( Q + ) ∋ u ( T u, u | ∂ Ω × R + , u | Ω ×{ } ) ∈ H , / , · ( Q + ) ∗ × H / , / · , · ( ∂ Ω × R + ) × L (Ω) , (4.66) is a linear isomorphism. ACKNOWLEDGEMENTS.
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